Calculator
Formula used
Interval to set-builder: [a, b] becomes {x ∈ D | a ≤ x ≤ b}. Open endpoints use the strict sign < instead.
Half-open intervals: [a, b) becomes {x ∈ D | a ≤ x < b}. Similarly, (a, b] and (a, b) follow the matching endpoint rules.
Finite arithmetic roster: {a, a + d, …, a + nd} becomes {x ∈ D | x = a + kd, 0 ≤ k ≤ n}.
Explicit finite roster: {a, b, c} becomes {x ∈ D | x = a or x = b or x = c} when no shorter repeating pattern exists.
How to use this calculator
- Select the conversion mode that matches your starting notation.
- Enter bounds, roster values, or a supported rule expression.
- Choose the domain so the calculator restricts valid members correctly.
- Set precision, preview size, and optional membership test values.
- Press Convert and analyze to see the result above the form.
- Use the CSV and PDF buttons to save the latest output.
Example data table
| Mode | Input | Domain | Output |
|---|---|---|---|
| Interval | [2, 7) | ℤ | {x ∈ ℤ | 2 ≤ x < 7} |
| Interval | (-1, 4] | ℝ | {x ∈ ℝ | -1 < x ≤ 4} |
| Roster | {1, 3, 5, 7} | ℤ | {x ∈ ℤ | 1 ≤ x ≤ 7 and (x - 1) mod 2 = 0} |
| Set-builder | {x ∈ ℕ | 3 ≤ x ≤ 6} | ℕ | [3, 6] with roster {3, 4, 5, 6} |
FAQs
1. What notation types can this page convert?
It converts bounded intervals, numeric rosters, and simple set-builder rules with inequalities or equality lists. It also creates previews, membership checks, and a number-line graph.
2. Why does the domain matter?
The domain decides which values qualify as members. For example, [2, 5] over ℤ gives integers only, while the same bounds over ℝ include every real value between them.
3. Does interval notation always match a roster exactly?
No. A scattered roster such as {1, 4, 9} is not a single interval. The calculator shows a bounding interval for context and keeps the exact rule in set-builder form.
4. What does the roster preview mean for real intervals?
Real intervals have infinitely many members. The preview therefore shows sample values instead of a complete list, while the exact condition stays in the interval and set-builder outputs.
5. Can I test whether values belong to the set?
Yes. Enter comma-separated test values in the membership field. The results table will show whether each value belongs to the converted set under the chosen domain.
6. What set-builder inputs are supported?
Use bounded forms like {x ∈ Z | 2 ≤ x < 8}, rule forms like {x in R | x >= -3 and x < 5}, or equality lists joined with “or”.
7. Why might the calculator return an empty set?
An empty set appears when the bounds and endpoint rules exclude every valid member. This often happens in discrete domains or when equal endpoints are made exclusive.
8. What do the CSV and PDF exports contain?
They contain the current conversion summary, the normalized rule, interval details, preview information, notes, and any membership-test outcomes shown in the result block.